Computing an integral by substitution is the reverse of the chain rule for computing the derivative. Substitution is intended to rewrite a complicated-looking integral involving the derivative of some component expression as another much simpler integral. For example, if $u=f(x)$ , then $\mathrm du=f'(x)\,\mathrm dx$ and

$\displaystyle\int f(x)f'(x)\,\mathrm dx=\int u\,\mathrm du$

###

Integration by parts is the reverse of the product rule for derivatives:

$(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)\implies f(x)g'(x)=(f(x)g(x))'-f'(x)g(x)$

Integrating both sides with respect to $x$ gives

$\displaystyle\int f(x)g'(x)\,\mathrm dx=\int(f(x)g(x))'\,\mathrm dx-\int f'(x)g(x)\,\mathrm dx$

$\displaystyle\int f(x)g'(x)\,\mathrm dx=f(x)g(x)-\int f'(x)g(x)\,\mathrm dx$

###

Personally, I think the best way to grasp the idea behind the two methods is to practice. You start to notice patterns to the point where knowing which is the "right" method to use becomes second nature.

4 0

3 years ago

By definition, future value is?

Yuki888 [10]

Future value is the value of an asset at a specific date. It measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, the rate of return; it is the present value multiplied by the accumulation function.

3 0

3 years ago

Read 2 more answers

Help me please i don't understand.

Misha Larkins [42]

The answer is b.....

4 0

3 years ago

Read 2 more answers